Efficiency. of infinite dimensional M- estimators

It is well-known that maximum likelihood estimators are asymptotically normal with covariance equal to the inverse Fisher information in smooth, finite dimensional parametric models. Thus they are asymptotically efficient. A similar phenomenon has been observed for certain infinite dimensional parameter spaces. We give a simple proof of efficiency, starting from a theorem on asymptotic normality of infinite dimensional M -estimators. The proof avoids the explicit calculation of the Fisher information. We also address Hadamard differentiability of the corresponding M -functionals.